In the PRHM scenario, the triple scalar couplings \(\lambda _{HHH}\) 
and \(\lambda _{HHS}\)  are strictly vanishing at tree-level. Therefore,
any process that it sensitive to these scalar triple couplings (like
\(pp\rightarrow HH@LHC14\)  and \(e^{-}e^{+}\rightarrow ZHH@ILC500\) 
as examples) would be fully triggered by radiative effects. Since
the radiative contributions to the scalar mixing (\(\Delta _{s_{\alpha }}\) )
are expected to be large and negative, one has consider a more precise
estimation for these triple couplings \(\lambda _{HHH}\)  and \(\lambda _{HHS}\) .
By considering the one-loop scalar mixing, these couplings should
be defined as the third derivatives of the one-loop effective potential
(REF ) following [1]}, which leads to
a significantly improved values of the (re-summed) couplings. In this
setup, each of these processes \(pp\rightarrow HH@LHC14\)  and \(e^{-}e^{+}\rightarrow ZHH@ILC500\) 
occurs via Feynman diagrams with the triple scalar couplings (\(\lambda _{HHH,HHS}\) )
and also via other diagrams without \(\lambda _{HHH,HHS}\) . Then, the
cross section can be divided as: (1) a contribution that involves
only \(\lambda _{HHH,HHS}\)  (\(\sigma _{\lambda }\) ) diagrams, (2) a pure
gauge couplings contribution (\(\sigma _{G}\) ); and (3) the interference
contribution (\(\sigma _{G\lambda }\) ). This makes the cross section
of both processes modified as
\(R(f)=\frac{\sigma (f)-\sigma _{SM}(f)}{\sigma _{SM}(f)}=\frac{\xi _{1}\sigma _{G}+\xi _{2}\sigma _{\lambda }+\xi _{3}\sigma _{G\lambda }}{\sigma _{G}+\sigma _{\lambda }+\sigma _{G\lambda }}-1,\) 
